Since Miller and Koblitz applied elliptic curves to cryptographic system in 1985 [3],[6], a lot of researchers have been interested in this field and various speedup techniques for the scalar multiplication have been developed. Recently, Gallant et al. published a method that accelerates the scalar multiplication and is applicable to a larger class of curves [4]. In the process of their method, they assumed the existence of a special pair of two short linearly independent vectors. Once a pair of such vectors exists, their decomposition method improves the efficiency of the scalar multiplication roughly about 50%. In this paper, we state and prove a necessary condition for the existence of a pair of desired vectors and we also present an algorithm to find them.